Definite Integrals: Definition, Properties, and Computation Methods | 定积分的定义、性质与计算方法

📚 Definite Integrals: Definition, Properties, and Computation Methods | 定积分的定义、性质与计算方法

The definite integral is one of the cornerstones of calculus, providing a rigorous way to compute areas, accumulated quantities, and net changes. This article introduces its definition, key properties, and practical computation techniques, aligned with A-level and international syllabus requirements.

定积分是微积分的核心内容之一，它为我们提供了一种严格计算面积、累积量和净变化量的方法。本文将围绕其定义、重要性质与常用计算方法展开，紧扣 A-level 及国际课程考试要求。

1. The Definition of the Definite Integral | 定积分的定义

The definite integral of a function f(x) from a to b is defined as the limit of a Riemann sum. Partition the interval [a, b] into n subintervals of equal width Δx = (b − a)/n, choose a sample point xᵢ* in each subinterval, and form the sum Σ f(xᵢ*) Δx. As n → ∞, if the limit exists, we write:

函数 f(x) 在区间 [a, b] 上的定积分定义为黎曼和的极限。将区间 [a, b] 分成 n 个等宽的子区间，宽度 Δx = (b − a)/n，在每个子区间内任取一点 xᵢ*，构造和式 Σ f(xᵢ*) Δx。当 n → ∞ 时，若该极限存在，则记作：

∫ₐᵇ f(x) dx = limₙ→∞ Σᵢ₌₁ⁿ f(xᵢ*) Δx

Here a is the lower limit, b is the upper limit, and dx indicates the variable of integration. If f(x) ≥ 0 on [a, b], the integral equals the area under the curve y = f(x) above the x-axis.

其中 a 称为积分下限，b 称为积分上限，dx 表示积分变量。若在 [a, b] 上 f(x) ≥ 0，则定积分等于曲线 y = f(x) 下方、x 轴上方的面积。

2. Existence and Integrability | 存在性与可积性

Not every function is integrable. A sufficient condition for integrability is continuity on the closed interval [a, b]. More generally, bounded functions with at most finitely many discontinuities are also Riemann integrable.

并非所有函数都可积。函数在闭区间 [a, b] 上连续是可积的一个充分条件。更一般地，有界且至多含有有限个间断点的函数也是黎曼可积的。

For example, f(x) = x² is integrable on [0, 1] because it is continuous. In contrast, a function that is unbounded on [a, b], such as f(x) = 1/x near x = 0, is not Riemann integrable on [0, 1].

例如，f(x) = x² 在 [0, 1] 上可积，因为它是连续的。相反，在 [a, b] 上无界的函数，如 f(x) = 1/x 在 x = 0 附近无界，就不是 [0, 1] 上的黎曼可积函数。

3. The Fundamental Theorem of Calculus | 微积分基本定理

The Fundamental Theorem of Calculus links differentiation and integration. If f is continuous on [a, b] and F is an antiderivative of f, then:

微积分基本定理将微分与积分联系起来。若 f 在 [a, b] 上连续，且 F 是 f 的一个原函数，则：

∫ₐᵇ f(x) dx = F(b) − F(a)

The theorem provides a powerful way to evaluate definite integrals without computing limits of sums. It also states that the derivative of an integral with a variable upper limit equals the integrand:

该定理提供了一种无需计算极限和式即可计算定积分的有力方法。它还指出，可变上限积分的导数等于被积函数：

d/dx ∫ₐˣ f(t) dt = f(x)

4. Basic Properties of Definite Integrals | 定积分的基本性质

The following properties are essential for simplifying and manipulating integrals.

以下性质对于化简和运算定积分至关重要。

Linearity: ∫ₐᵇ [c f(x) + d g(x)] dx = c ∫ₐᵇ f(x) dx + d ∫ₐᵇ g(x) dx

线性：∫ₐᵇ [c f(x) + d g(x)] dx = c ∫ₐᵇ f(x) dx + d ∫ₐᵇ g(x) dx
Additivity over intervals: If a ≤ c ≤ b, then ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx

区间可加性：若 a ≤ c ≤ b，则 ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx
Zero-length interval: ∫ₐᵃ f(x) dx = 0

零长度区间：∫ₐᵃ f(x) dx = 0
Reversing limits: ∫ᵦₐ f(x) dx = − ∫ₐᵇ f(x) dx

交换上下限：∫ᵦₐ f(x) dx = − ∫ₐᵇ f(x) dx
Comparison: If f(x) ≤ g(x) on [a, b], then ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx

比较性质：若在 [a, b] 上 f(x) ≤ g(x)，则 ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx

5. The Mean Value Theorem for Integrals | 积分中值定理

If f is continuous on [a, b], there exists at least one number c in (a, b) such that:

若 f 在 [a, b] 上连续，则至少存在一点 c ∈ (a, b)，使得：

∫ₐᵇ f(x) dx = f(c)(b − a)

The value f(c) is called the average value of f on [a, b], defined as:

称 f(c) 为 f 在 [a, b] 上的平均值，其定义为：

f(c) = (1/(b − a)) ∫ₐᵇ f(x) dx

This theorem is useful when estimating integrals and solving problems involving average values.

该定理常用于估计积分值以及处理涉及平均值的题目。

6. Computation Using Antiderivatives | 利用原函数计算定积分

The most common method is to find an antiderivative and apply the Fundamental Theorem of Calculus. Standard formulas include:

最常用的方法是寻找原函数并应用微积分基本定理。基本公式包括：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1
∫ eˣ dx = eˣ + C

∫ eˣ dx = eˣ + C
∫ (1/x) dx = ln|x| + C

∫ (1/x) dx = ln|x| + C
∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C

∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C

For example, ∫₀¹ x² dx = [x³/3]₀¹ = 1/3 − 0 = 1/3.

例如，∫₀¹ x² dx = [x³/3]₀¹ = 1/3 − 0 = 1/3。

7. Integration by Substitution | 换元积分法

Substitution is used when the integrand contains a composite function. If x = g(u), then dx = g′(u) du, and the limits must be changed accordingly:

换元法适用于被积函数含有复合函数的情形。若 x = g(u)，则 dx = g′(u) du，且定积分的上下限需要相应改变：

∫ₐᵇ f(g(x)) g′(x) dx = ∫_{g(a)}^{g(b)} f(u) du

Example: Evaluate ∫₀² 2x(x²+1)³ dx. Let u = x²+1, then du = 2x dx. When x = 0, u = 1; when x = 2, u = 5. Hence:

例：计算 ∫₀² 2x(x²+1)³ dx。令 u = x²+1，则 du = 2x dx。当 x = 0 时，u = 1；当 x = 2 时，u = 5。因此：

∫₁⁵ u³ du = [u⁴/4]₁⁵ = (625 − 1)/4 = 156

Remember to change the limits of integration when using substitution on a definite integral.

注意：对定积分使用换元法时，一定要同时更换积分上下限。

8. Integration by Parts | 分部积分法

Integration by parts is derived from the product rule. For definite integrals:

分部积分法由乘积法则推导而来。对于定积分：

∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du

The choice of u and dv is critical. A common guideline is to choose u as a function that simplifies upon differentiation (e.g., x, x², ln x) and dv as a part that is easy to integrate.

选择 u 和 dv 至关重要。一般原则是：令 u 为求导后可以简化的函数（如 x、x²、ln x），而 dv 为容易积分的部分。

Example: Evaluate ∫₀¹ x eˣ dx. Let u = x, dv = eˣ dx, so du = dx, v = eˣ. Then:

例：计算 ∫₀¹ x eˣ dx。令 u = x，dv = eˣ dx，则 du = dx，v = eˣ。于是：

[x eˣ]₀¹ − ∫₀¹ eˣ dx = e − (e − 1) = 1

9. Symmetry and Even/Odd Functions | 对称性与奇偶函数

Exploiting symmetry can greatly simplify integrals over symmetric intervals.

利用对称性可以大大简化对称区间上的积分。

If f is even, f(−x) = f(x), then:

若 f 为偶函数，即 f(−x) = f(x)，则：

∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx

If f is odd, f(−x) = −f(x), then:

若 f 为奇函数，即 f(−x) = −f(x)，则：

∫₋ₐᵃ f(x) dx = 0

For example, ∫₋₁¹ x³ dx = 0 (odd), while ∫₋₁¹ x² dx = 2 ∫₀¹ x² dx = 2/3.

例如，∫₋₁¹ x³ dx = 0（奇函数），而 ∫₋₁¹ x² dx = 2 ∫₀¹ x² dx = 2/3。

10. Area Between Curves | 曲线间的面积

The definite integral can be used to find the area between two curves. If f(x) ≥ g(x) on [a, b], the area bounded by the curves is:

定积分可用于求两条曲线之间的面积。若在 [a, b] 上 f(x) ≥ g(x)，则两曲线所围成的面积为：

A = ∫ₐᵇ [f(x) − g(x)] dx

When curves intersect, find the intersection points first to determine the integration limits. For example, the area between y = x and y = x² on [0, 1] is:

当曲线相交时，需先求交点以确定积分限。例如，y = x 与 y = x² 在 [0, 1] 之间的面积为：

∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/6

11. Numerical Methods: Trapezoidal and Simpson’s Rules | 数值方法：梯形法则与辛普森法则

When an antiderivative is difficult or impossible to find, numerical integration is used. The trapezoidal rule approximates the integral by summing areas of trapezoids:

当原函数难以或无法求出时，可使用数值积分。梯形法则通过求和梯形面积来近似积分：

∫ₐᵇ f(x) dx ≈ (Δx/2) [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]

Simpson’s rule uses quadratic approximations and is more accurate:

辛普森法则使用二次函数近似，精度更高：

∫ₐᵇ f(x) dx ≈ (Δx/3) [f(x₀) + 4f(x₁) + 2f(x₂) + … + 4f(xₙ₋₁) + f(xₙ)]

where n must be even for Simpson’s rule. In A-level exams, questions may ask you to estimate an integral from a table of values using these rules.

其中辛普森法则要求 n 为偶数。在 A-level 考试中，题目可能要求你根据一组函数值用这些法则估算积分值。

12. Common Traps and Exam Tips | 常见易错点与应试提示

Mastering these details will help you avoid careless mistakes in exams.

掌握以下细节有助于避免考试中的粗心错误。

Always apply the limits correctly when using substitution; forgetting to change limits is a common error.

使用换元法时务必正确更换上下限，忘记更换上下限是常见错误。
Check the sign of the integrand when computing areas. If f(x) is negative, ∫ₐᵇ f(x) dx gives a negative value; to find area, take the absolute value.

求面积时注意被积函数的符号。若 f(x) 为负，则 ∫ₐᵇ f(x) dx 为负值；求面积需取绝对值。
For integration by parts, choose u and dv wisely; sometimes repeated integration by parts is needed.

用分部积分法时谨慎选择 u 和 dv；有时需要多次分部积分。
Remember the property ∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx when reversing limits.

交换上下限时牢记 ∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx。
For even/odd functions on symmetric intervals, simplify before integrating.

在对称区间上遇到奇偶函数，先化简再积分。

Practice with past paper questions to build fluency and confidence.

多练历年真题，提升熟练度与信心。

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Definite Integrals: Definition, Properties, and Computation Methods | 定积分的定义、性质与计算方法